Investigation of the Group Lasso Algorithm for Sound Field Reproduction: Comparison with the Lasso and Elastic-net Algorithms

The reproduction of a sound field measured using a microphone array is an active topic of research. To this end, loudspeaker and microphone arrays are used. Classical methods rely on spatial transforms (such as spherical Fourier transform for Ambisonics) or pressure matching using least-mean-square formulation. For both methods, all the reproduction sources (i.e. loudspeakers) will typically be activated. Although this can provide a reduced or minimized reproduction error evaluated at the microphone array, it is not necessarily the most useful solution for listening purposes. Indeed, the fact that all reproduction sources are concurrently active can potentially lead to a blurry spatial image (an example would be the common front-back confusion). To solve that potential limitation, the lasso and elastic-net algorithms were recently studied in order to favor the sparsity of the reproduction source signals. In these studies, it was shown that sparsity can indeed lead to a sharper source distribution at the cost of reduced physical accuracy of the reproduced sound field. In this paper, the group lasso is investigated to alleviate such potential limitations of the lasso, where the "group" refer to groups of reproduction sources. The aim of the group lasso is to provide sparsity at the group level and continuous smooth solution inside groups. In the recent literature, many simple or detailed algorithms have been proposed for the real-valued group lasso without a consensual position from the community. For sound field reproduction, a simple algorithmic implementation is proposed as an interpretation of the group lasso for complex quantities. Simulation results in free field show that several of the limitations of the lasso and elastic-net algorithms are solved. Potentials and current limitations of the group lasso are discussed. Based on the reported investigation, future research openings include: the possibility of overlapping groups and the algorithmic implementation.